The Dirichlet heat trace for domains with curved corners

Shi Zhuo Looi; David Sher

Paper

The Dirichlet heat trace for domains with curved corners

Abstract

We study the short-time asymptotics of the Dirichlet heat trace on planar curvilinear polygons. For such domains we show that the coefficient of

in the expansion splits into a boundary integral of

and a sum of local corner contributions, one for each vertex. Each curved corner contribution depends only on the interior angle

and on the limiting curvatures

on the adjacent sides. Using a conformal model and a parametrix construction on the sector heat space, we express this contribution in the form

, where

is given by a Hadamard finite part of an explicit trace over the exact sector. For right-angled corners we compute

and obtain a closed formula for the

coefficient. As an application we extend a previous result in the literature by showing that any admissible curvilinear polygon that is Dirichlet isospectral to a polygon must itself be a polygon with straight sides.